A fast convergence density evolution algorithm for optimal rate LDPC codes in BEC

Abstract

We derive a new fast convergent Density Evolution algorithm for finding optimal rate Low-Density Parity-Check (LDPC) codes used over the binary erasure channel (BEC). The fast convergence property comes from the modified Density Evolution (DE), a numerical method for analyzing the behavior of iterative decoding convergence of a LDPC code. We have used the method of [16] for designing of a LDPC code with optimal rate. This has been done for a given parity check node degree distribution, erasure probability and specified DE constraint. The fast behavior of DE and found optimal rate with this method compare with the previous DE constraint.

Overhead minimization of online codes

“…Solving the overhead minimization problem based on the Linear Programming (LP) method is possible but some points of the constraints are not considered. For more details consider (8), which has too many constraint points for different continuous variable of x. Now, consider a set of x i such as {x i } N i 1 and each component belongs to interval [δ, 1].…”

“…Without loss of generality, we consider that the length of each block is one. Erasure channel losses a fraction δ of the packets and it is well-known that a code with rate R 1 − δ can achieve the capacity [3][4][5][6][7][8]. Let us define like the original online code that each check block with degree i is given by the XOR of i message bits, which are selected randomly.…”

“…where for typical online codes we have a b 0. δparameter is the maximum erasing probability of bits in transmitting and the related capacity of the channel is 1 − δ. It is clear that (8) guarantees that for all possible degree distributions such as α(x) and β(x), the decoder converges. Now, without loss of generality consider that the right degree distribution is fixed and we want to minimize the overhead of the online code to achieve a better performance.…”

“…According to the recent and new advances in error control coding theory, especially with using the regular [1] and irregular [2][3][4][5][6][7][8] form of Low Density Parity Check (LDPC) codes, efficient error correction schemes can be found. By using the LDPC codes and their well-known decoders such as Belief Propagation (BP) algorithm, one can achieve the whole channel capacity of the Binary Erasure Channel (BEC) [9][10][11][12] and approach to a rate, which is very close to channel capacity of Binary Symmetric Channel (BSC) [10], or Additive White Gaussian Noise Channel (AWGNC) [13].…”

Raptor-like Coded Broadcasting for Efficient V2X Communications

“…Specifically, we set the degree distribution of check nodes as ρ(x) = ∑ D cm j=2 ρ j x j−1 and the degree distribution of bit nodes as λ(x) = ∑ D vm i=2 λ i x i−1 , where D vm and D cm represent the possible maximum degree value of the bit nodes and check nodes, respectively. Then, the coding rate during transmission is calculated as [26]…”